| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 10.8·5-s − 36·6-s + 229.·7-s − 64·8-s + 81·9-s + 43.2·10-s + 144·12-s − 903.·13-s − 916.·14-s − 97.2·15-s + 256·16-s − 763.·17-s − 324·18-s + 2.27e3·19-s − 172.·20-s + 2.06e3·21-s + 2.82e3·23-s − 576·24-s − 3.00e3·25-s + 3.61e3·26-s + 729·27-s + 3.66e3·28-s − 6.97e3·29-s + 389.·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.193·5-s − 0.408·6-s + 1.76·7-s − 0.353·8-s + 0.333·9-s + 0.136·10-s + 0.288·12-s − 1.48·13-s − 1.25·14-s − 0.111·15-s + 0.250·16-s − 0.640·17-s − 0.235·18-s + 1.44·19-s − 0.0966·20-s + 1.02·21-s + 1.11·23-s − 0.204·24-s − 0.962·25-s + 1.04·26-s + 0.192·27-s + 0.883·28-s − 1.53·29-s + 0.0789·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 10.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 229.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 903.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 763.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.27e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.97e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.05e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.10e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.02e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.02e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 8.75e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 8.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.43e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.40e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.36e4T + 8.58e9T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.161667962661323085534901614496, −8.340303420602241645533831930100, −7.41969989203563387440328169429, −7.27013834187553397306208500916, −5.42225462700969921889162790460, −4.77390203900029248851853754727, −3.44676001976992204153638130289, −2.15418180920939093248805437174, −1.49173352109158241995036798686, 0,
1.49173352109158241995036798686, 2.15418180920939093248805437174, 3.44676001976992204153638130289, 4.77390203900029248851853754727, 5.42225462700969921889162790460, 7.27013834187553397306208500916, 7.41969989203563387440328169429, 8.340303420602241645533831930100, 9.161667962661323085534901614496
